We study perturbative renormalization of the composite operators in the $T\bar T$-deformed two-dimensional free field theories. The pattern of renormalization for the stress-energy tensor is different in the massive and massless cases. While in the latter case the canonical stress tensor is not renormalized up to high order in the perturbative expansion, in the massive theory there are induced counterterms at linear order. For a massless theory our results match the general formula derived recently in [1].
We propose a general path-integral definition of two-dimensional quantum field theories deformed by an integrable, irrelevant vector operator constructed from the components of the stress tensor and those of a $U(1)$ current. The deformed theory is obtained by coupling the original QFT to a flat dynamical gauge field and "half" a flat dynamical vielbein. The resulting partition function is shown to satisfy a geometric flow equation, which perfectly reproduces the flow equations for the deformed energy levels that were previously derived in the literature. The S-matrix of the deformed QFT diffe…
We consider the $γ$-deformed $\mathcal{N}=4$ SYM in the double scaling limit of large imaginary twists and small coupling, which discards the gauge fields and retains only certain Yukawa and scalar interactions with three arbitrary effective couplings. In the 't~Hooft limit, these 4D theories are integrable, with most of the correlators being conformal such that the whole arsenal of AdS/CFT integrability remains applicable. In particular, for one non-zero effective coupling, we obtain a QFT of two complex scalars with a chiral, quartic interaction. The BMN vacuum anomalous dimension is dominat…
In a recent paper it was shown that the response of an integrable QFT under variation of the Unruh temperature can be computed from a S-matrix preserving deformation of the form factor approach. We give explicit expressions for the deformed two-particle formfactors for various integrable models: The Sine-Gordon and SU(2) Thirring model, several perturbed minimal CFTs and the real coupling affine Toda series. A uniform pattern is found to emerge when both the S-matrix and the deformed form factors are expressed in terms Barnes' multi-periodic functions.
We study strongly coupled mass-deformed-CFT on a fixed de Sitter spacetime in three dimensions via holography. We elucidate the global causal structure of the four-dimensional spacetime dual to the de Sitter invariant vacuum state. The conformal boundaries of de Sitter appear as spacelike defects sourced by the mass deformation, which extend into the bulk as curvature singularities in AdS. We compute all one- and two-point functions of the deformed-CFT stress tensor and a scalar operator order-by-order in the mass deformation for a simple holographic model. These correlation functions admit a …
We consider a general (beyond $T\bar T$) deformation of the 2D $O(N+1)$ $σ$-model by the irrelevant dimension-four operators. The theory deformed in this most general way is not integrable, and the $S$-matrix loses its factorization properties. We perform the all-order summation of the leading infrared logs for the $2 \to 2$ scattering amplitude and provide the exact result for the $2 \to 2$ $S$-matrix in the leading logarithmic approximation. These results can provide us with new insights into the properties of the theories deformed by irrelevant operators more general than the $T\bar T$ defo…
Various holographic set-ups in string theory suggest the existence of non-local, UV complete two-dimensional QFTs that possess Virasoro symmetry, in spite of their non-locality. We argue that $J\bar T$-deformed CFTs are the first concrete realisation of such "non-local CFTs", through a detailed analysis of their classical and quantum symmetry algebra. Classically, the symmetries consist of an infinite set of left-moving conformal and affine $U(1)$ transformations that generate a Witt-Kac-Moody algebra, as well as a set of non-local, field-dependent generalizations of right-moving conformal and…
We study (2,2s+1) RSOS lattice models deformed by the current-current operator. Solving the deformed Bethe ansatz equations for the model in the regime III we find explicit expressions for the ground state energy as well as for the energy, the momentum and the scattering matrix of the breather-like excitations. In the scaling limit we get an additional confirmation to the proposal that the bi-local deformations are a lattice analogue of the TT perturbations of integrable massive QFTs proposed by Smirnov and Zamolodchikov.
$J\bar T$-deformed CFTs provide an interesting example of non-local, yet UV-complete two-dimensional QFTs that are entirely solvable. They have been recently shown to possess an infinite set of symmetries, which are a continuous deformation of the Virasoro-Kac-Moody symmetries of the seed CFT. In this article, we put forth a definition of primary operators in $J\bar T$-deformed CFTs on a cylinder, which are singled out by having CFT-like momentum-space commutation relations with the symmetry generators in the decompatification limit. We show -- based on results we first derive for the case of …
The holographic duality between cascading gauge theory and type IIB supergravity on warped deformed conifold with fluxes reveals exotic thermal phases with nonzero expectation values of certain operators, persistent to high temperatures. These phases, in the limit of vanishing the strong coupling scale of the cascading gauge theory, would realize thermal ordered conformal phases in relativistic QFTs. We find that the dual Klebanov-Strassler and Klebanov-Tseytlin black branes in this limit are outside the regime of the supergravity approximation, rendering the construction of such conformal ord…
We examine the renormalized free energy of the free Dirac fermion and the free scalar on a (2+1)-dimensional geometry $\mathbb{R} \times Σ$, with $Σ$ having spherical topology and prescribed area. Using heat kernel methods, we perturbatively compute this energy when $Σ$ is a small deformation of the round sphere, finding that at any temperature the round sphere is a local maximum. At low temperature the free energy difference is due to the Casimir effect. We then numerically compute this free energy for a class of large axisymmetric deformations, providing evidence that the round sphere global…
Composite bosons (or quasibosons), as recently proven, are realizable by deformed oscillators and due to that can be effectively treated as particles of nonstandard statistics (deformed bosons). This enables us to study quasiboson states and their inter-component entanglement aspects using the well developed formalism of deformed oscillators. We prove that the internal entanglement characteristics for single two-component quasiboson are determined completely by the parameter(s) of deformation. The bipartite entanglement characteristics are generalized and calculated for arbitrary multi-quasibo…
Composite structure of particles somewhat modifies their statistics, compared to the pure Bose- or Fermi-ones. The spin-statistics theorem, so, is not valid anymore. Say, $π$-mesons, excitons, Cooper pairs are not ideal bosons, and, likewise, baryons are not pure fermions. In our preceding papers, we studied bipartite composite boson (i.e. quasiboson) systems via a realization by deformed oscillators. Therein, the interconstituent entanglement characteristics such as entanglement entropy and purity were found in terms of the parameter of deformation. Herein, we perform an analogous study of co…
Geometric realization of stress-tensor deformed field theory
Center for Theoretical Physics and College of Physics, Jilin University（吉林大学理论物理中心及物理学院）；Institute of Fundamental Physics and Quantum Technology & School of Physical Science and Technology, Ningbo University（宁波大学基础物理与量子技术研究所及物理科学与技术学院）；Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理研究所（阿尔伯特·爱因斯坦研究所））
We study finite size effects for composite operators in the SU(2) sector of the superconformal beta-deformed N=4 SYM theory. In particular we concentrate on the spectrum of one single magnon. Since in this theory one-impurity states are non BPS we compute their anomalous dimensions including wrapping contributions up to four loops and discuss higher order effects.
We study two families of composite twisted Ramond fields (made by products of two operators) in the $\cal {N}=(4,4)$ supersymmetric D1-D5 SCFT$_2$ deformed by a marginal modulus operator away from its $(T^4)^N/ S_N$ free orbifold point. We construct the large-$N$ contributions to the four-point functions with two composite operators and two deformation fields. These functions allow us to derive short-distance OPE limits and to calculate the anomalous dimensions of the composite operators. We demonstrate that one can distinguish two sets of composite Ramond states with twists $m_1$ and $m_2$: p…
The concept of quasi-bosons or composite bosons (like mesons, excitons etc.) has a wide range of potential physical applications. Even composed of two pure fermions, the quasi-boson creation and annihilation operators satisfy non-standard commutation relations. It is natural to try to realize the quasi-boson operators by the operators of deformed (nonlinear) oscillator, the latter constituting widely studied field of modern quantum physics. In this paper, it is proved that the deformed oscillators which realize quasi-boson operators in a consistent way really exist. The conditions for such rea…
We consider the deformed Bose gas model with the deformation structure function that is the combination of a q-deformation and a quadratically polynomial deformation. Such a choice of the unifying deformation structure function enables us to describe the interacting gas of composite (two-fermionic or two-bosonic) bosons. Using the relevant generalization of the Jackson derivative, we derive a two-parametric expression for the total number of particles, from which the deformed virial expansion of the equation of state is obtained. The latter is interpreted as the virial expansion for the effect…
We consider the relativistic phase space coordinates (x_μ,p_μ) as composite, described by functions of the primary pair of twistor coordinates. It appears that if twistor coordinates are canonicaly quantized the composite space-time coordinates are becoming noncommutative. We obtain deformed Heisenberg algebra which in order to be closed should be enlarged by the Pauli-Lubanski four-vector components. We further comment on star-product quantization of derived algebraic structures which permit to introduce spin-extended deformed Heisenberg algebra.
We investigate various perturbative properties of the deformed N=4 SYM theory. We carry out a three-loops calculation of the chiral matter superfield propagator and derive the condition on the couplings for maintaining finiteness at this order. We compute the 2-, 3- and 4-point functions of composite operators of dimension 2 at two loops. We identify all the scalar operators (chiral and non-chiral) of bare dimension 4 with vanishing one-loop anomalous dimension. We compute some 2- and 3-point functions of these operators at two loops and argue that the observed finite corrections cannot be abs…
Using $Γ_{\pm}(z) $ vertex operators of the $c=1$ two dimensional conformal field theory, we give a 2d-quantum field theoretical derivation of the conjectured d- dimensional MacMahon function G$_{d}(q) $. We interpret this function G$_{d}(q) $ as a $(d+1) $- point correlation function $\mathcal{G}_{d+1}(z_{0},...,z_{d}) $ of some local vertex operators $\mathcal{O}%_{j}(z_{j}) $. We determine these operators and show that they are particular composites of q-deformed hierarchical vertex operators $% Γ_{\pm}^{(p)}$, with a positive integer p. In agreement with literature's results, we find that …
We study the $SU(2)_k/U(1)$-parafermion model perturbed by its first thermal operator. By formulating the theory in terms of a (perturbed) fermionic coset model we show that the model is equivalent to interacting WZW fields modulo free fields. In this scheme, the order and disorder operators of the $Z_k$ parafermion theory are constructed as gauge invariant composites. We find that the theory presents a duality symmetry that interchanges the roles of the spin and dual spin operators. For two particular values of the coupling constant we find that the theory recovers conformal invariance and th…
For composite systems made of $N$ different particles living in a space characterized by the same deformed Heisenberg algebra, but with different deformation parameters, we define the total momentum and the center-of-mass position to first order in the deformation parameters. Such operators satisfy the deformed algebra with new effective deformation parameters. As a consequence, a two-particle system can be reduced to a one-particle problem for the internal motion. As an example, the correction to the hydrogen atom $n$S energy levels is re-evaluated. Comparison with high-precision experimental…
The point-splitting regularization technique for composite operators is discussed in connection with anomaly calculation. We present a pedagogical and self-contained review of the topic with an emphasis on the technical details. We also develop simple algebraic tools to handle the path ordered exponential insertions used within the covariant and non-covariant version of the point-splitting method. The method is then applied to the calculation of the chiral, vector, trace, translation and Lorentz anomalies within diverse versions of the point-splitting regularization and a connection between th…
Bipartite composite boson (quasiboson) systems, which admit realization in terms of deformed oscillators, were considered in our previous paper from the viewpoint of entanglement characteristics. These characteristics, including entanglement entropy and purity, were expressed through the relevant deformation parameter for different quasibosonic states. On the other hand, it is of interest to present the entanglement entropy and likewise the purity as function of energy for those states. In this work, the corresponding dependencies are found for different states of composite bosons realized by …
It has been recently shown that the deformation of an arbitrary two-dimensional conformal field theory by the composite irrelevant operator $T \bar T$, built from the components of the stress tensor, is solvable; in particular, the finite-size spectrum of the deformed theory can be obtained from that of the original CFT through a universal formula. We study a similarly universal, Lorentz-breaking deformation of two-dimensional CFTs that posess a conserved $U(1)$ current, $J$. The deformation takes the schematic form $J \bar T$ and is interesting because it preserves an $SL(2,\mathbb{R}) \times…
Composite bosons, here called {\it quasibosons} (e.g. mesons, excitons, etc.), occur in various physical situations. Quasibosons differ from bosons or fermions as their creation and annihilation operators obey non-standard commutation relations, even for the "fermion+fermion" composites. Our aim is to realize the operator algebra of quasibosons composed of two fermions or two q-fermions (q-deformed fermions) by the respective operators of deformed oscillators, the widely studied objects. For this, the restrictions on quasiboson creation/annihilation operators and on the deformed oscillator (de…
We formulate $λ$-deformed $σ$-models as QFTs in the upper-half plane. For different boundary conditions we compute correlation functions of currents and primary operators, exactly in the deformation parameter $λ$ and for large values of the level $k$ of the underlying WZW model. To perform our computations we use either conformal perturbation theory in association with Cardy's doubling trick, as well as meromorphicity arguments and a non-perturbative symmetry in the parameter space $(λ,k)$, or standard QFT techniques based on the free field expansion of the $σ$-model action, with the free fiel…
It was noticed many years ago, in the framework of massless RG flows, that the irrelevant composite operator $T \bar{T}$, built with the components of the energy-momentum tensor, enjoys very special properties in 2D quantum field theories, and can be regarded as a peculiar kind of integrable perturbation. Novel interesting features of this operator have recently emerged from the study of effective string theory models.In this paper we study further properties of this distinguished perturbation. We discuss how it affects the energy levels and one-point functions of a general 2D QFT in finite vo…
We continue our study of $λ$-deformed $σ$-models by setting up a $1/k$ perturbative expansion around the free field point for cosets, in particular for the $λ$-deformed $SU(2)/U(1)$ coset CFT. We construct an interacting field theory in which all deformation effects are manifestly encoded in the interaction vertices. Using this we reproduce the known $β$-function and the anomalous dimension of the composite operator perturbing away from the conformal point. We introduce the $λ$-dressed parafermions which have an essential Wilson-like phase in their expressions. Subsequently, we compute their a…
Recently, a non-local yet possibly UV-complete quantum field theory has been constructed by deforming a two-dimensional CFT by the composite operator $J \bar T$, where $J$ is a chiral $U(1)$ current and $\bar T$ is a component of the stress tensor. Assuming the original CFT was a holographic CFT, we work out the holographic dual of its $J \bar T$ deformation. We find that the dual spacetime is still AdS$_3$, but with modified boundary conditions that mix the metric and the Chern-Simons gauge field dual to the $U(1)$ current. We show that when the coefficient of the chiral anomaly for $J$ vanis…
We consider conformal field theories in 1+1 dimensions with W-algebra symmetries, deformed by a chemical potential μfor the spin-three current. We show that the order μ^2 correction to the Re'nyi and entanglement entropies of a single interval in the deformed theory, on the infinite spatial line and at finite temperature, is universal. The correction is completely determined by the operator product expansion of two spin-three currents, and by the expectation values of the stress tensor, its descendants and its composites, evaluated on the n-sheeted Riemann surface branched along the interval. …
The $\text{T}\bar{\text{T}}$-deformed classical Lagrangian of a 2D Lorentz invariant theory can be derived from the original one, perturbed only at first order by the bare $\text{T}\bar{\text{T}}$ composite field, through a field-dependent change of coordinates. Considering, as an example, the nonlinear Schrödinger (NLS) model with generic potential, we apply this idea to non-relativistic models. The form of the deformed Lagrangian contains a square-root and is similar but different from that for relativistic bosons. We study the deformed bright, grey and Peregrine's soliton solutions. Contrar…
Boundary flow and geometric realization in holographic $T\bar T$-deformed BCFT
Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）
We investigate the behaviour of two-dimensional quantum field theories with $\mathcal{N}=(0,2)$ supersymmetry under a deformation induced by the `$T\bar{T}$' composite operator. We show that the deforming operator can be defined by a point-splitting regularisation in such a way as to preserve $\mathcal{N}=(0,2)$ supersymmetry. As an example of this construction, we work out the deformation of a free $\mathcal{N}=(0,2)$ theory and compare to that induced by the Noether stress-energy tensor. Finally, we show that the $\mathcal{N}=(0,2)$ supersymmetric deformed action actually possesses $\mathcal…
Non-commutative Quantum Mechanics in 3D is investigated in the framework of the abelian Drinfeld twist which deforms a given Hopf algebra while preserving its Hopf algebra structure. Composite operators (of coordinates and momenta) entering the Hamiltonian have to be reinterpreted as primitive elements of a dynamical Lie algebra which could be either finite (for the harmonic oscillator) or infinite (in the general case). The deformed brackets of the deformed angular momenta close the so(3) algebra. On the other hand, undeformed rotationally invariant operators can become, under deformation, an…
Local products of fields deformed by the so-called Yang--Mills gradient flow become renormalized composite operators. This fact has been utilized to construct a correctly normalized conserved energy--momentum tensor in the lattice formulation of the pure Yang--Mills theory. In the present paper, this construction is further generalized for vector-like gauge theories containing fermions.
In this thesis, we investigate properties of the one-parameter $β$- and the three-parameter $γ_i$-deformed descendents of $\mathcal{N}=4$ SYM theory. We find additional multi-trace interactions that must be included in the deformed theories to prevent persistent divergences in observables. For the $γ_i$-deformation, we show in an explicit Feynman-diagrammatic calculation that these interactions have running coupling constants which spoil the conformal invariance of the quantised theory, even in the 't Hooft limit. We show that the running coupling constant in principle also affect the anomalou…
The conformal symmetry algebra in 2D (Diff($S^{1}$)$\oplus$Diff($S^{1}$)) is shown to be related to its ultra/non-relativistic version (BMS$_{3}$$\approx$GCA$_{2}$) through a nonlinear map of the generators, without any sort of limiting process. For a generic classical CFT$_{2}$, the BMS$_{3}$ generators then emerge as composites built out from the chiral (holomorphic) components of the stress-energy tensor, $T$ and $\bar{T}$, closing in the Poisson brackets at equal time slices. Nevertheless, supertranslation generators do not span Noetherian symmetries. BMS$_{3}$ becomes a bona fide symmetry…
We consider a one-parameter family of composite fields -- bi-linear in the components of the stress-energy tensor -- which generalise the $\mathrm{T}\bar{\mathrm{T}}$ operator to arbitrary space-time dimension $d\geq 2$. We show that they induce a deformation of the classical action which is equivalent -- at the level of the dynamics -- to a field-dependent modification of the background metric tensor according to a specific flow equation. Even though the starting point is the flat space, the deformed metric is generally curved for any $d>2$, thus implying that the corresponding deformation ca…
Surprising links between the deformation of 2D quantum field theories induced by the composite $\textrm{T} \bar{\textrm{T}}$ operator, effective string models and the $AdS/$CFT correspondence, have recently emerged. The purpose of this article is to discuss various classical aspects related to the deformation of 2D interacting field theories. Special attention is given to the sin(h)-Gordon model, for which we were able to construct the $\textrm{T} \bar{\textrm{T}}$-deformed Lax pair. We consider the Lax pair formulation to be the first essential step toward a more satisfactory geometrical inte…
